coarsening coefficient matrixes (A2h, A4h...) for geometric multigrid method in 2-D/3-D

Question

I am learning about multigrid methods from the textbook section 6.3 Multigrid Methods, which shows a geometric multigrid algorithm for 1-D examples in detail, including how to build restriction/interpolation matrices (i.e., eqs.1, 3) and how to get a coarse coefficient matrix (eq. 6).

I realize that these matrices and way of coarsening the coefficient matrix are important for multigrid methods.

But for the 2-D case, only the ways of getting restriction and interpolation matrices are shown in the book, not the coefficient matrix for coarse grid (i.e., $A_{2h}$, $A_{4h}$, etc.).

This missing link is barrier for me to go further to learn multigird methods.

Can anybody help me?

Answer 1

For geometric multigrid, the $\mathbf A_{2h}$ and $\mathbf A_{4h}$ (etc) matrices are just discretizations of the same PDE on coarser grids.

For instance, if your original $\mathbf A_{h}$ was a finite difference approximation of the laplacian on a 64 point grid, then $\mathbf A_{2h}$ would be a finite difference approximation of the laplacian on a 32 point grid, etc. You should also see some sort of relationship where $\mathbf A_{2h} = \mathbf R \mathbf A_h \mathbf R^T$ (maybe to within a constant? I forget).

In algebraic multigrid, only the latter relationship really applies. There is no "fine grid" or "coarse grid", just the matrices themselves, and the restriction operator itself uniquely defines the coarsened problem(s).